On packing spheres into containers (about Kepler's finite sphere packing problem)

In an Euclidean $d$-space, the container problem asks to pack $n$ equally sized spheres into a minimal dilate of a fixed container. If the container is a smooth convex body and $d\geq 2$ we show that solutions to the container problem can not have a ``simple structure'' for large $n$. By this we in particular find that there exist arbitrary small $r>0$, such that packings in a smooth, 3-dimensional convex body, with a maximum number of spheres of radius $r$, are necessarily not hexagonal close packings. This contradicts Kepler's famous statement that the cubic or hexagonal close packing ``will be the tightest possible, so that in no other arrangement more spheres could be packed into the same container''.Comment: 13 pages, 2 figures; v2: major revision, extended result, simplified and clarified proo

Separation with restricted families of sets

Given a finite $n$-element set $X$, a family of subsets ${\mathcal F}\subset 2^X$ is said to separate $X$ if any two elements of $X$ are separated by at least one member of $\mathcal F$. It is shown that if $|\mathcal F|>2^{n-1}$, then one can select $\lceil\log n\rceil+1$ members of $\mathcal F$ that separate $X$. If $|\mathcal F|\ge \alpha 2^n$ for some $0<\alpha<1/2$, then $\log n+O(\log\frac1{\alpha}\log\log\frac1{\alpha})$ members of $\mathcal F$ are always sufficient to separate all pairs of elements of $X$ that are separated by some member of $\mathcal F$. This result is generalized to simultaneous separation in several sets. Analogous questions on separation by families of bounded Vapnik-Chervonenkis dimension and separation of point sets in ${\mathbb{R}}^d$ by convex sets are also considered.Comment: 13 page

Discrete isometry groups of symmetric spaces

This survey is based on a series of lectures that we gave at MSRI in Spring 2015 and on a series of papers, mostly written jointly with Joan Porti. Our goal here is to: 1. Describe a class of discrete subgroups $\Gamma<G$ of higher rank semisimple Lie groups, which exhibit some "rank 1 behavior". 2. Give different characterizations of the subclass of Anosov subgroups, which generalize convex-cocompact subgroups of rank 1 Lie groups, in terms of various equivalent dynamical and geometric properties (such as asymptotically embedded, RCA, Morse, URU). 3. Discuss the topological dynamics of discrete subgroups $\Gamma$ on flag manifolds associated to $G$ and Finsler compactifications of associated symmetric spaces $X=G/K$. Find domains of proper discontinuity and use them to construct natural bordifications and compactifications of the locally symmetric spaces $X/\Gamma$.Comment: 77 page

The quantum measurement problem enhanced

The quantum measurement problem as formalised by Bassi and Ghirardi [Phys. Lett. A 275 (2000)373] without taking recourse to sharp apparatus observables is extended to cover impure initial states.Comment: 6 page